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MathGroup Archive 2006

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Re: Known recursion link to Hermite polynomials not solved in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69478] Re: Known recursion link to Hermite polynomials not solved in Mathematica
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 14 Sep 2006 06:54:45 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <ee647q$5nb$1@smc.vnet.net>

In article <ee647q$5nb$1 at smc.vnet.net>,
 Roger Bagula <rlbagula at sbcglobal.net> wrote:

> A000898 in Sloan's OEIS:
> 
> a[0] = 1; a[1] = 2;
> a[n_] := a[n] = 2*(a[n - 1] + (n - 1)*a[n - 2])
> b = Table[a[n], {n, 0, 30}]
> 
> {1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 
> 30710464,
> 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 
> 2396286830080,
> 17138748412928, 125336396368896, 936222729254912, 7136574106003456,
> 55466948299223040, 439216305474605056, 3540846129311916032,
> 29042507532354084864, 242209013788927803392, 2052713434324976189440,
> 17669131640829909368832, 154395642472508437725184}
> 
> Table[HermiteH[n, I]*I^n, {n, 0, 30}]
> 
> {1, -2, 6, -20, 76, -312, 1384, -6512,
>    32400, -168992, 921184, -5222208, 30710464, -186753920, 1171979904,
> -7573069568, 50305536256, -342949298688, 2396286830080, -17138748412928,
> 125336396368896, -936222729254912,
>    7136574106003456, -55466948299223040, 439216305474605056,
> -3540846129311916032, 29042507532354084864, -242209013788927803392,
> 2052713434324976189440, -17669131640829909368832, 154395642472508437725184}
> 
> Yet Mathematica refuses to give that solution ( using Bob Hanlon's solver):
> f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == 2*( a[n - 1] + (n -  
> 1)*a[n - 2]), a[0] == 1, a[1] == 2}, a[n], n][[1]] // FullSimplify]

However, Mathematica can find this solution using

  <<DiscreteMath`

  ExponentialGeneratingFunction[{a[0] == 1, a[1] == 2, 
    a[n] == 2 (a[n - 1] + (n - 1) a[n - 2])}, a[n], n, z]

and, since Exp[-z^2 + 2 z s] is the ExponentialGeneratingFunction for 
HermiteH[n, s], the connection is clear.

> My reason for wanting a solution is my own Hermite related recursion 
> A121966:
> a[0] = 1; a[1] = 2;
> a[n_] := a[n] = a[n - 1] - (n - 1)*a[n - 2]
> Table[a[n], {n, 0, 30}]
> 
> {1, 2, 1, -3, -6, 6, 36, 0, -252, -252, 2016, 4536, -17640, -72072,
>   157248, 1166256, -1192464, -19852560, 419328, 357765408, 349798176,
> -6805509984, -14151271680, 135569947968, 461049196608, -2792629554624,
> -14318859469824, 58289508950400, 444898714635648, -1187207535975552,
> -14089270260409344}
> 
> Which I think has some simple HermiteH type solution.

I don't think so. If you try

  ExponentialGeneratingFunction[{a[0] == 1, a[1] == 2, 
    a[n] == a[n - 1] - (n - 1) a[n - 2]}, a[n], n, z]

you will find that the solution involves Exp[-(z - 1)^2 / 2] -- which 
leads to HermiteH -- but that this multiplies

  Erfi[(z - 1)/Sqrt[2]]

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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